Why every number to the power of 0 is equal to 1?

Question

I’m too dumb to get it

In: Mathematics

Answer 1

Let’s use 2 as the base of exponent for simplicity.

Exponents are naturally defined only for positive integers. Exponent tells you how many 2’s you multiply together. 2^3 for example is 2 multiplied by itself 3 times, 2 * 2 * 2.

If you take two numbers like 2^3 and 2^2, and multiply them together, you have 2 multiplied by itself 3 times, and then 2 times. (2x2x2)x(2×2) = 2x2x2x2x2 = 2^5. We can just add 2 and 3 together to know how many 2’s we are multiplying together.

In math speak, this means 2^n * 2^m = 2^(n+m)

Now, we are almost done. You see, exponents are supposed to be positive, but actually this equation seems to work just fine for all integers. So let’s try putting in 0.

2^3 * 2^0

According to our rule, our cool formula, this should equal 2^(3+0) = 2^3. But we don’t really know yet what this 2^0 means. Let’s study it. We know what 2^3 is, it’s 2 * 2 * 2 = 8. So let’s try to solve for 2^0

8 * 2^0 = 8

Divide both sides by 8:

1 * 2^0 = 1

2^0 = 1

Well that was easy. We can use 2^0 when using our cool formula above, and it tells us that the value we should assign to 2^0 is 1. So we can just do that.

In a very similar way, you can figure out what values we should use for negative or fractional exponents.

So in short, we noticed that the natural definition, “how many 2’s we multiply together” obeyed this really neat equation. But then we noticed that we can actually use that equation for values that don’t seem to make sense. And as it turns out, it works.

In mathematics this kind of thing is very common. We notice a pattern, and then we start to apply this pattern onto new things, things where it might not make sense. And if it works out, well, that’s pretty cool. We can start using it for new things and find new patterns, and so on.

In some cases, it’s also possible to find out that everything breaks if you try some extensions or other such ideas.

There’s this very strong sense among mathematicians that some extensions are “natural”. They flow out of the initial definitions almost by themselves. While one could try to be more rigorous about it, I think it’s helpful to try to see these things through the lens of aesthetics. Does this thing look pretty, does it feel right? You eventually want to prove that things actually work and all that, but it often starts with this feeling of something being natural